Question 277975: PLEASE HELP!!
If an earthquake has a magnitude of 4.2 on the Richter scale, what is the magnitude on the Richter scale of an earthquake that has an intensity 20 times greater?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the richter scale is based on the log of 10.
richter scale of 0 is equivalent to the log of an earthquake with a magnitude of 10^0.
richter scale of 1 is equivalent to the log of an earthquake with a magnitude of 10^1.
richter scale of 2 is equivalent to the log of an earthquake with a magnitude of 10^2.
richter scale of 3 is equivalent to the log of an earthquake with a magnitude of 10^3.
richter scale of 4 is equivalent to the log of an earthquake with a magnitude of 10^4.
richter scale of 5 is equivalent to the log of an earthquake with a magnitude of 10^5.
richter scale of 6 is equivalent to the log of an earthquake with a magnitude of 10^6.
richter scale of 7 is equivalent to the log of an earthquake with a magnitude of 10^7.
richter scale of 8 is equivalent to the log of an earthquake with a magnitude of 10^8.
richter scale of 9 is equivalent to the log of an earthquake with a magnitude of 10^9.
richter scale of 10 is equivalent to the log of an earthquake with a magnitude of 10^10.
since the richter scale is based on the log to the base 10, this means that each succeeding number on the richter scale is 10 * bigger than the previous number.
richter scale of 0 = log(10^0) = log(1) = earthquake with a magnitude of 1
richter scale of 1 = log(10^1) = log(10) = earthquake with a magnitude of 10
richter scale of 2 = log(10^2) = log(100) = earthquake with a magnitude of 100
richter scale of 3 = log(10^3) = log(1000) = earthquake with a magnitude of 1000
etc.
now to your problem.
problem states that earthquake has a magnitude of 4.2 on the richter scale.
this means that the log of x to the base 10 = 4.2
by the laws of logarithms, that is true if and only if:
x = 10^4.2
now you can use your calculator to find 10^4.2 and your equation becomes:
x = 15848.93192
that means that an earthquake with a magnitude of 15848.93192 measures 4.2 on the richter scale.
for an earthquake to be 20 * more powerful than that, x would have to be equal to 20 * 15848.93192 as shown below:
x = 20 * 15848.93192 = 316978.6385
so you have x = 316978.6385 and you take the log of that to the base 10 to get:
log(316978.6385) = 5.501029996
this is true because 10^5.501029996 = 316978.6385.
your answer is that an earthquake with an intensity of 20 times greater would measure 5.501029996 on the richter scale.
you could have solved this a little differently as shown below:
if an earthquake measures 4.2 on the richter scale, this means that:
4.2 = log(x) where x is the magnitude of the earthquake.
for the earthquake to be 20 times as powerful, then the new measurement on the richter scale becomes:
y = log(20*x)
by the laws of logarithms, this becomes:
y = log(20) + log(x)
since log(x) = 4.2, this becomes:
y = log(20) + 4.2
using your calculator to find log(20), this becomes:
y = 1.301029996 + 4.2 which becomes:
y = 5.501029996
that's the same answer we got before.
we originally found x and then took 20 times that number.
this time we did not find x, but used the laws of logarithms.
either way we got the same answer.
the basic rules of logarithms are:
y = log(b,x) if and only if b^y = x
log(a*x) = log(x) + log(a)
log(x/a) = log(x) - log(a)
log(x^a) = a * log(x)
either way, your answer is:
an earthquake with a magnitude 20 * greater than an earthquake that measures 4.2 on the richter scale is an earthquake that measures 5.501029996 on the richter scale.
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